On the mean square average of special values of L-functions

Let χ be a Dirichlet character and L(s,χ) be its L-function. Using weighted averages of Gauss and Ramanujan sums, we find exact formulas involving Jordan?s and Euler?s totient function for the mean square average of L(1,χ) when χ ranges over all odd characters modulo k and L(2,χ) when χ ranges over all even characters modulo k. In principle, using our method, it is always possible to find the mean square average of L(r,χ) if χ and r?1 have the same parity and χ ranges over all odd (or even) characters modulo k, though the required calculations become formidable when r?3. Consequently, we see that for almost all odd characters modulo k, |L(1,χ)|<Φ(k), where Φ(x) is any function monotonically tending to infinity.     

The eighth moment of Dirichlet L-functions

We prove an asymptotic for the eighth moment of Dirichlet L-functions averaged over primitive characters χ modulo q  , over all moduli q?Q$q?Q$ and with a short average on the critical line, conditionally on GRH. We derive the analogous result for the fourth moment of Dirichlet twists of GL(2)$\mathit{GL}(2)$L-functions. Our results match the moment conjectures in the literature; in particular, the constant 24 024 appears as a factor in the leading order term of the eighth moment.     

Euler numbers congruences and Dirichlet L-functions

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In this paper we apply Yamamoto's Theorem [Y. Yamamoto, Dirichlet series with periodic coefficients, in: Proc. Intern. Sympos. “Algebraic Number Theory”, Kyoto, 1976, JSPS, Tokyo, 1977, pp. 275-289] to find the residue modulo a prime power of the linear combination of Dirichlet L-function values L(s,χ) at positive integral arguments s such that s and χ are of the same parity, in terms of Euler numbers, whereby we obtain the finite expressions for short interval character sums. The results obtained generalize the previous results pertaining to the congruences modulo a prime power of the class numbers as the special case of s=1.

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The L-functions of twisted Witt extensions

The L-function of a non-degenerate twisted Witt extension is proved to be a polynomial. Its Newton polygon is proved to lie above the Hodge polygon of that extension. And the Newton polygons of the Gauss-Heilbronn sums are explicitly determined, generalizing the Stickelberger theorem.     

On the vanishing of Selmer groups for elliptic curves over ring class fields

Let E/Q be an elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let Hc be the ring class field of K of conductor c prime to ND with Galois group Gc over K. Fix a complex character χ of Gc. Our main result is that if LK(E,χ,1)≠0 then Selp(E/Hc)χ_⊗W=0 for all but finitely many primes p, where Selp(E/Hc) is the p-Selmer group of E over Hc and W is a suitable finite extension of Zp containing the values of χ. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a χ-twisted version of the Birch and Swinnerton-Dyer conjecture for E over Hc (Bertolini and Darmon) and of the vanishing of Selp(E/K) for almost all p (Kolyvagin) in the case of analytic rank zero.     

On linear congruence relations for Kubota-Leopoldt 2-adic L-functions

Our purpose in the paper is to find the most general linear congruence relation of the Hardy-Williams type for linear combinations of special values of Kubota-Leopoldt 2-adic L-functions L₂(k,χω^1−k) with k running over any finite subset of not necessarily consisting of consecutive integers (see Acta Arith. 47 (1986) 263; Publ. Math. Fac. Sci. Besançon, Théorie des Nombres, 1995/1996; Publ. Math. Debrecen 56 (2000) 677 and cf. Mathematics and Its Applications, Vol. 511, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000). If k runs over finite subsets of consisting of consecutive integers see Compositio Math. 111 (1998) 289; Publ. Math. Debrecen 56 (2000) 677; Hardy and Williams, 1986; Compositio Math. 75 (1990) 271; Acta Arith. 71 (1995) 273; 52 (1989) 147; J. Number Theory 34 (1990) 362. In order to obtain the most general congruences of this type we make use of divisibility properties of the generalized Vandermonde determinants obtained in Spie? et al. (Divisibility properties of generalized Vandermonde and Cauchy determinants, Preprint 627, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 2002). This allows us to simplify our main Theorem 2 and obtain Theorem 3 where the most general form of the linear congruence relation is given.     

Average frobenius distributions for elliptic curves with 3-torsion

In this paper, we examine the Lang-Trotter conjecture for elliptic curves which possess rational 3-torsion points. We prove that if one averages over all such elliptic curves then one obtains an asymptotic similar to the one predicted by Lang and Trotter.     

On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

Complete classification of torsion of elliptic curves over quadratic cyclotomic fields

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In a previous paper Najman (in press) [9], the author examined the possible torsions of an elliptic curve over the quadratic fields Q(i) and . Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over Q(i) and , we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves over Q(i) and .

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Linnik-type problems for automorphic L-functions

Let m?2 be an integer, and π an irreducible unitary cuspidal representation for GLm(AQ), whose attached automorphic L-function is denoted by L(s,π). Let be the sequence of coefficients in the Dirichlet series expression of L(s,π) in the half-plane Rs>1. It is proved in this paper that, if π is such that the sequence is real, then there are infinitely many sign changes in the sequence , and the first sign change occurs at some , where Qπ is the conductor of π, and the implied constant depends only on m and ε. This generalizes the previous results for GL₂. A result of the same quality is also established for , the sequence of coefficients in the Dirichlet series expression of in the half-plane Rs>1.     

On elliptic units and p-adic Galois representations attached to elliptic curves

Let K be a quadratic imaginary number field with discriminant D_K≠-3,-4 and class number one. Fix a prime p?7 which is not ramified in K and write h_p for the class number of the ray class field of K of conductor p. Given an elliptic curve A/K with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that if then the image of a certain deformation of is “as big as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Z_p). The proof rests on the theory of Siegel functions and elliptic units as developed by Kubert, Lang and Robert.     

Positivity of certain functions associated with analysis on elliptic surfaces

In this paper, we study functions of one variable that are called boundary terms of two-dimensional zeta integrals established in recent works of Ivan Fesenko?s two-dimensional adelic analysis attached to arithmetic elliptic surfaces. It is known that the positivity of the fourth log derivatives of boundary terms around the origin is a sufficient condition for the Riemann hypothesis of Hasse-Weil L-functions of elliptic curves. We show that such positivity is also a necessary condition under some reasonable technical assumptions.     

The vanishing order of certain Hecke L-functions of imaginary quadratic fields

Let −D<−4 denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of exists. Let d be a fundamental discriminant prime to D. Let 2k−1 be an odd natural number prime to the class number of . Let χ be the twist of the (2k−1)th power of a canonical Hecke character of by the Kronecker's symbol . It is proved that the vanishing order of the Hecke L-function L(s,χ) at its central point s=k is determined by its root number when , where the constant implied in the symbol ? depends only on k and ?, and is effective for L-functions with root number −1.     

On the behaviour of p-adic L-functions

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Let Lp(s,χ) denote a Leopoldt-Kubota p-adic L-function, where p>2 and χ is a nonprincipal even character of the first kind. The aim of this article is to study how the values assumed by this function depend on the Iwasawa λ-invariant associated to χ. Assuming that λ?p−1, it turns out that Lp(s,χ) behaves, in some sense, like a polynomial of degree λ. The results lead to congruences of a new type for (generalized) Bernoulli numbers.

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q-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functions

The aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define q-analogue of the Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, and q-Genocchi type l-function. We define partial zeta function. By using this function, we construct p-adic interpolation functions which interpolate generalized q-Genocchi numbers at negative integers. We also define p-adic meromorphic functions on C_p. Furthermore, we construct new generating functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.     

Evaluation of double L-values

In this paper, a theorem of Zagier concerning double zeta evaluations is generalized to the double L-values. In addition, fast computation of the double L-values is demonstrated, extending the method of Crandall. The PARI commands are available electronically.     

Constructing one-parameter families of elliptic curves with moderate rank

We give several new constructions for moderate rank elliptic curves over Q(T). In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over Q using polynomials of degree two in T. While our method generates linearly independent points, we are able to show the rank is exactly 6 without having to verify the points are independent. The method generalizes; however, the higher rank surfaces are not rational, and we need to check that the constructed points are linearly independent.     

Thue equations and torsion groups of elliptic curves

A new characterization of rational torsion subgroups of elliptic curves is found, for points of order greater than 4, through the existence of solution for systems of Thue equations.     

Central values of L-functions over CM fields

We prove an explicit formula for the central values of certain Rankin L-functions. These L-functions are the L-functions attached to Hilbert newforms over a totally real field F, twisted by unitary Hecke characters of a totally imaginary quadratic extension of F. This formula generalizes our former result on L-functions twisted by finite CM characters.